As an object falls inside a liquid, it experiences drag and lift forces, and torques depending on the shape of the object. In the case of a sphere both the flow field and drag has been both theoretically calculated and epxerimentally determined for a wide range of terminal velocities. The key parameter that determines the flow field and the drag on a sphere is the Reynolds number: $$ Re = \frac{\rho L U}{\mu}, $$ were \(L\) and \(U\) are the characteristic length and velocity scales, and \(\mu\) and \(\rho\) the dynamic viscosity and density of the fluid. In the case of very low \(Re\), the flow field is symmetric (see Fig. 1(a)) and the drag force is given by: $$ D = 6\pi \mu R U, $$ with \(R\) the radius of the sphere. In this research we want to investigate the sendimentation of toroidal objects in the Stokes regime, where \(Re<<1\). It has been shown that the total drag on a torus in this regime depends also the aspect ratio of the torus, \(\xi=R_0/a_0\) (see Fig. 3(a)).

Fig. 1: (a) A schematic and the flow field of a falling sphere. (b) A schematic and (c) the apparatus of the PIV setup.

To do this, we want to investigate the flow field around a torus. We do this by peforming particle image velocimetry (PIV). PIV relies on optical inhomogeneities in the fluid which can be correlated over time to extract the local velocity fields. We introduce the optical inhomogeneities by mixing \(16\) \(\mu m\) polystyrene particles in our fluid. To ensure that we are in the Stokes regime, we increase the viscosity of our fluid by using a mixture of glycerol and water. We image the optical inhomogeneities in a plane by using a laser sheet as shown in figures 1(b,c).

Video: A sinking sphere.		Fig. 2: The correspodning flow field around the sinking sphere.

We quantify the validity of our setup by performing experiments with sinking spheres (see Video 1). By performing PIV on this video, we can extract the flow field in the reference frame of the sphere (see Fig. 2), which indeed matches the flow field around a sphere (see Fig. 1(a)).

Fig. 3: (a) A schematic of the geometrical parameters of a torus. (b) A toroidal hydrogel.

The toroidal objects are hydrogels. They are made in the lab by generating troidal droplets of a solution of acrylic acid and BIS, a cross linker. After the torus is made, the polymerica reaction is catalyzed using UV radiation. The hydrogels are then cleaned and equilibrated in a solution with the same consistency as the tank. An example of such a hydrogel is shown in figure 3(b).

Some of our results show that tori tend to rotate as the fall in the fluid and become horizontal, as shown in video 2, indicating that a net torque is exerted on the torus causing it to rotate. An example of a torus falling through the viscous can be seen in video 3.

Video 2: A sinking torus rotating.	Video 3: Optical inhomogeneities around a falling torus.

References:

[1] Utada, A. S., Fernandez-Nieves, A., Stone, H. A., Weitz, D. A., Dripping to jetting transitions in coflowing liquid streams, Physical Review Letters 99, 094502 (2007).

[2] Sauret, A., Cheung Shum, H., Forced generation of simple and double emulsions in all-aqueous systems. Applied Physics Letters 100, 154106 (2012).